## How a Message Becomes Corrupted in Being Passed Along

When a story is told and retold in being passed from person to person it often gets morphed by degrees into something unrecognisable, so that the origin of the story does not recognise it all.
Suppose that someone says 'yes' to a question, and their answer gets relayed along a chain of individuals. Each individual has a probability
$p$
of passing on the message 'yes' correctly, and a probability
$1-p$
of passing on the exactly opposite message 'no'
The probabilities that each person will pass on the message as they received it or not are summarised below.
 Message Received\Message Passed Yes No Yes $p$ $1-p$ No $1-p$ $p$
The transition matrix is
$T=\left( \begin{array}{cc} p & 1-p \\ 1-p & p \end{array} \right)$
.
Suppose after many relays of the message, we have the probability vector
$(x,y)$
, then passing one more time will not change this so solve
$(x,y ) \left( \begin{array}{cc} p & 1-p \\ 1-p & p \end{array} \right)=(x,y)$
.
With
$x+y=1$
(letting
$x, \; y$
be probabilities of the initial message being either 'yes' or 'no').
We have
$xp+y(1-p)=x \righarrow y(1-p)=x(1-p) \rightarrow x=y$
.
Hence the probability of the message passed along eventually becoming 'yes' approaches 1/2 and the probability of the message eventually becoming 'no' approaches 1/2, whatever the initial message and whatever the value of
$p$
.